- Fixed point theorem
In

mathematics , a**fixed point theorem**is a result saying that a function "F" will have at least one fixed point (a point "x" for which "F"("x") = "x"), under some conditions on "F" that can be stated in general terms. Results of this kind are amongst the most generally useful in mathematics.**Fixed point theorem in analysis**The

Banach fixed point theorem gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point.By contrast, the

Brouwer fixed point theorem is a non-constructive result: it says that any continuous function from the closedunit ball in "n"-dimensionalEuclidean space to itself must have a fixed point, but it doesn't describe how to find the fixed point (See alsoSperner's lemma ).For example, the

cosine function is continuous in [-1,1] and maps it into [-1, 1] , and thus must have a fixed point. This is clear when examining a sketched graph of the cosine function; the fixed point occurs where the cosine curve $y=cos(x)$ intersects the line $y=x$. Numerically, the fixed point is approximately $x=0.73908513321516$ (thus $x=cos(x)$).The

Lefschetz fixed-point theorem (and the Nielsen fixed-point theorem) fromalgebraic topology is notable because it gives, in some sense, a way to count fixed points.There are a number of generalisations to

Banach space s and further; these are applied in PDE theory. Seefixed point theorems in infinite-dimensional spaces .The

collage theorem infractal compression proves that, for many images, there exists a relatively small description of a function that, when iteratively applied to any starting image, rapidly converges on the desired image.**Fixed point theorems in discrete mathematics and theoretical computer science**The

Knaster-Tarski theorem is somewhat removed from analysis and does not deal with continuous functions. It states that any order-preserving function on acomplete lattice has a fixed point, and indeed a "smallest" fixed point. See alsoBourbaki-Witt theorem .A common theme in

lambda calculus is to find fixed points of given lambda expressions. Every lambda expression has a fixed point, and afixed point combinator is a "function" which takes as input a lambda expression and produces as output a fixed point of that expression. An important fixed point combinator is theY combinator used to give recursive definitions.In

denotational semantics of programming languages, a special case of theKnaster-Tarski theorem is used to establish the semantics of recursive definitions. While the fixed point theorem is applied to the "same" function (from a logical point of view), the development of the theory is quite different.The same definition of recursive function can be given, in

computability theory , by applyingKleene's recursion theorem . These results are not equivalent theorems; theKnaster-Tarski theorem is a much stronger result than what is used indenotational semantics . [*"The foundations of program verification", 2nd edition, Jacques Loeckx and Kurt Sieber, John Wiley & Sons, ISBN 0 471 91282 4, Chapter 4; theorem 4.24, page 83, is what is used in denotational semantics, while*] However, in light of theKnaster-Tarski theorem is given to prove as exercise 4.3-5 on page 90.Church-Turing thesis their intuitive meaning is the same: a recursive function can be described as the least fixed point of a certain functional, mapping functions to functions.The above technique of iterating a function to find a fixed point can also be used in

set theory ; thefixed-point lemma for normal functions states that any continuous strictly increasing function from ordinals to ordinals has one (and indeed many) fixed points.Every

closure operator on aposet has many fixed points; these are the "closed elements" with respect to the closure operator, and they are the main reason the closure operator was defined in the first place.**ee also***

Atiyah–Bott fixed-point theorem

*Borel fixed-point theorem

*Brouwer fixed point theorem

*Caristi fixed point theorem

*Diagonal lemma

*Fixed point property

*Injective metric space

*Kakutani fixed-point theorem

*Kleene fixpoint theorem

*Woods Hole fixed-point theorem

*Topological degree theory **Notes****References***cite book

author = Agarwal, Ravi P.; Meehan, Maria; O'Regan, Donal

title = Fixed Point Theory and Applications

year = 2001

publisher = Cambridge University Press

id = ISBN 0-521-80250-4*cite book

author = Border, Kim C.

title = Fixed Point Theorems with Applications to Economics and Game Theory

year = 1989

publisher = Cambridge University Press

id = ISBN 0-521-38808-2*cite book

author = Brown, R. F. (Ed.)

title = Fixed Point Theory and Its Applications

year = 1988

publisher = American Mathematical Society

id = ISBN 0-8218-5080-6*cite book

author = Dugundji, James; Granas, Andrzej

title = Fixed Point Theory

year = 2003

publisher = Springer-Verlag

id = ISBN 0-387-00173-5*cite book

author = Kirk, William A.; Goebel, Kazimierz

title = Topics in Metric Fixed Point Theory

year = 1990

publisher = Cambridge University Press

id = ISBN 0-521-38289-0

*cite book

author = Kirk, William A.; Khamsi, Mohamed A.

title = An Introduction to Metric Spaces and Fixed Point Theory

year = 2001

publisher = John Wiley, New York.

id = ISBN 978-0-471-41825-2

*cite book

author = Kirk, William A.; Sims, Brailey

title = Handbook of Metric Fixed Point Theory

year = 2001

publisher = Springer-Verlag

id = ISBN 0-7923-7073-2*cite book

author = Šaškin, Jurij A; Minachin, Viktor; Mackey, George W.

title = Fixed Points

year = 1991

publisher = American Mathematical Society

id = ISBN 0-8218-9000-X**Links**[

*http://www.math-linux.com/spip.php?article60 Fixed Point Method*]

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